Black hole evaporation |
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A brief analysis of the mathematical results of Hawking radiation. | |||
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Black hole evaporation. | |||
The Hawking temperature T of a Schwarzschild (nonrotating, uncharged) black hole with mass m is given by the equation (in geometrized units) [reference 1] | |||
T = hbar/(8 pi k m). | equation 1 | ||
In conventional units (which we use here), this would be written | |||
T = (hbar c3)/(8 pi G k m). | equation 2 | ||
The emission of this energy results in an energy decrease of the black
hole, and thus a loss in its mass. What period of time tau will it
take for a black hole of mass mu to evaporate completely? A black hole with mass m has a Schwarzschild radius |
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r = 2 G m/c2 | equation 3 | ||
and thus an area of | |||
A = 4 pi r2 | equation 4 | ||
A = 16 pi G2 m2/c4. | equation 5 | ||
Hawking radiation would have a power P related to the hole's area A and its temperature T by the blackbody power law (with e = 1), | |||
P = sigma A T4 | equation 6 | ||
P = (sigma hbar4 c8)/(256 pi3 G2 k4 m2) | equation 7 | ||
or more conveniently, | |||
P = K/m2 | equation 8 | ||
where K == (sigma hbar4 c8)/(256 pi3 G2 k4) = 3.563 x 1032 W kg2. Given that the power of the Hawking radiation is the rate of energy loss of the hole, we can write | |||
P = -dE/dt. | equation 9 | ||
Since the total energy E of the hole is related to its mass m by Einstein's mass-energy formula, | |||
E = m c2 | equation 10 | ||
we can then rewrite P = -dE/dt as | |||
P = -(d/dt) (m c2) | equation 11 | ||
P = -c2 dm/dt. | equation 12 | ||
We can then equate this to our above expression for the power, P = K/m2, and find | |||
-c2 dm/dt = K/m2. | equation 13 | ||
This differential equation is separable, and we can write | |||
m2 dm = -K/c2 dt. | equation 14 | ||
Integrating over m from mu (the initial mass of the hole) to zero (complete evaporation), and over t from zero to tau, we find that | |||
tau = c2/(3 K) mu3. | equation 15 | ||
That is, the evaporation time of the hole is proportional to the cube
of its mass.
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References. | |||
1. Black holes, white dwarfs, and neutron stars: The physics of compact objects Stuart L. Shapiro, Saul A. Teukolsky p. 366 Wiley-Interscience; 1983 |
reference 1 | ||
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